4. MEASUREMENT OF INTRACLUSTER MAGNETIC FIELDS

As stressed above, the presence of diffuse and extended synchrotron
emission in galaxy clusters indicates the existence of weak magnetic
fields in the cluster volume. Different possibilities for their origin
have been proposed which are reviewed by
Dolag et al. (2008)
- Chapter 15, this volume. Radio observations of galaxy clusters allow
us to measure intracluster magnetic fields and test the different
theories on their origin, as reviewed by
Carilli &
Taylor (2002)
and
Govoni &
Feretti (2004).
In the following the main methods to study magnetic
field intensity and, eventually, structure are summarised.

In the optically thin case, the total monochromatic emissivity
J() from a set of
relativistic electrons in a magnetic field
B depends on a) the magnetic field strength, b)
the energy distribution of the electrons, which is usually assumed to
be a power law (Eq. 1), and c) the pitch angle between the
electron velocity and the magnetic field direction
()

(3)

where =
( - 1) / 2 is the
spectral index of the
synchrotron spectrum 4.

Synchrotron emission from diffuse and extended radio sources can give
us a direct measure for the intensity of cluster magnetic fields if
the relativistic electron flux is measured or constrained. That can be
achieved, for example, if Compton-produced X-ray (and gamma-ray)
emission was detected simultaneously (see Sect. 4.2).
In the case
of polarised radio emission, we can also get an indication of the
projected magnetic field orientation and its degree of ordering. To
break the degeneracy between magnetic field strength and electron
density (Eq. 3), and to obtain a measure for cluster
magnetic fields from the observed luminosity of radio sources, it is
typically assumed that the energy density of the relativistic plasma
within a radio source is minimum

(4)

where UB is the energy density in magnetic fields, and
Uel and Upr are the energy in
electrons and in protons respectively. The energy in the heavy particles
(protons) is considered to be related to the electron energy

(5)

The value of k depends on the mechanism of
(re-)acceleration of electrons, whose physical details, as seen above,
are still unknown. A typical value of k = 1 is adopted for halo and
relic sources. Another important assumption of this method relates to
the value of the filling factor,
, i.e. the fraction of the
source volume V occupied by magnetic field and relativistic
particles. The energy density in magnetic field is given by

(6)

It is usually considered that particles and magnetic fields
occupy the entire volume, i.e.
= 1. It can be derived
easily that the condition of minimum energy is obtained when the
contributions of cosmic rays and magnetic fields is approximately equal

(7)

This is the so-called classical equipartition assumption,
which allows us to estimate the magnetic field of a radio source from
its radio luminosity L (see
Pacholczyk
1970
for a rigorous derivation)

(8)

In the standard approach presented above, L is the
observed synchrotron luminosity between two fixed frequencies
1
and 2 (usually
1 = 10 MHz and
2 = 100 GHz). In
this way, however, the integration limits are variable in terms of the
energy of the radiating electrons, since, based on Eq. 3, electron
energies corresponding to
1 and
2 depend on
magnetic field
values. This point is particularly relevant for the lower limit, owing
to the power-law shape of the electron energy distribution and to the
expected presence of low energy electrons in radio
halos/relics. Alternatively, it has been suggested to derive
equipartition quantities by integrating the electron luminosity over
an energy range (min -
max)
(Brunetti et
al. 1997,
Beck & Krause
2005).
It can be shown that, for min << max
and > 0.5, the new
expression for the equipartition magnetic field is

(9)

where Beq is the equipartition magnetic field expressed
in Gauss derived through Eq. 8. Typically, for Beq ~
µG, min ~ 100 and ~ 0.75 - 1, this new
approach gives magnetic field values 2 to 5 times larger than the
standard method.

Estimates of equipartition fields on scales as large as ~1 Mpc
give magnetic field intensities in the range 0.1-1 µG. As we
have seen, these estimates are based on several assumptions both on
different physical properties of the radio emitting region (e.g. the
filling factor and the
ratio between electron and proton
energies k), and on the condition of minimum energy of the observed
relativistic plasma. Since the validity of these assumptions is not
obvious, one has to be aware of the uncertainties and thus of the
limits inherent to the equipartition determination of magnetic fields.

As reviewed by
Rephaeli et
al. (2008)
- Chapter 5, this volume, 3K
microwave background photons can be subject to Compton scattering by
electrons in the cluster volume. If the presence of thermal particles
in the ICM results in a distortion of the Cosmic Microwave Background
(CMB) spectrum well known as "Sunyaev-Zel'dovich effect"
(Sunyaev &
Zel'dovich 1972),
non-thermal hard X-ray (HXR) photons are produced
via Compton scattering by the same cosmic rays that are responsible
for the synchrotron emission observed at radio wavelengths. Compton
scattering increases the frequency of the incoming photon through

Non-thermal HXR emission from galaxy clusters due to Compton
scattering of CMB photons was predicted more than 30 years ago (e.g.
Rephaeli 1977)
and has now been detected in several systems
(Rephaeli et
al. 2008
- Chapter 5, this volume;
Fusco-Femiano et
al. 2007
and references therein). Alternative
interpretations to explain the detected non-thermal X-ray emission
have been proposed in the literature
(Blasi &
Colafrancesco 1999,
Enßlin et
al. 1999,
Blasi 2000,
Dogiel 2000,
Sarazin &
Kempner 2000).
However, these hypotheses seem to be
ruled out by energetic considerations, because of the well known
inefficiency of the proposed non-thermal Bremsstrahlung (NTB)
mechanism. NTB emission of keV regime photons with some power P
immediately imply about 105 times larger power to be
dissipated in
the plasma that seems to be unrealistic in a quasi-steady model
(Petrosian 2001,
2003).
For a more detailed treatment of the
origin of HXR emission from galaxy clusters, see the review by
Petrosian et
al. (2008)
- Chapter 10, this volume.

The detection of non-thermal HXR and radio emission, produced by the
same population of relativistic electrons, allows us to estimate
unambiguously the volume-averaged intracluster magnetic
field. Following the exact derivations by
Blumenthal
& Gould (1970),
the equations for the synchrotron flux fsyn at the
frequency R
and the Compton X-ray flux fC at the frequency
X are

(11)

(12)

Here h is the Planck constant, V is the volume of the
source and the filling
factor, DL is the luminosity distance
of the source, B the magnetic field strength, T the radiation
temperature of the CMB, r0 the classical electron
radius (or Thomson scattering length), N0 and
are the amplitude and
the spectral index of the electron energy distribution
(Eq. 1). The values of the functions
a() and
F() for different
values of can be found
in
Blumenthal
& Gould (1970).
The field B can thus be estimated directly from these equations

Faraday rotation analysis of radio sources in the background or in the
galaxy clusters themselves is one of the key techniques used to obtain
information on the cluster magnetic fields. The presence of a
magnetised plasma between an observer and a radio source changes the
properties of the polarised emission from the radio source. Therefore
information on cluster magnetic fields along the line-of-sight can be
determined, in conjunction with X-ray observations of the hot gas,
through the analysis of the Rotation Measure (RM) of radio sources (e.g.
Burn 1966).

The polarised synchrotron radiation coming from radio galaxies
undergoes the following rotation of the plane of polarisation as it
passes through the magnetised and ionised intracluster medium

(14)

where Int is
the intrinsic polarisation angle, and
Obs() is the polarisation angle
observed at a wavelength
. The RM is related to
the thermal electron
density (ne), the magnetic field along the line-of-sight
(B||), and the path-length (L) through the
intracluster medium according to

(15)

Polarised radio galaxies can be mapped at several
frequencies to produce, by fitting Eq. 14, detailed RM images.
Once the contribution of our Galaxy is subtracted, the RM should be
dominated by the contribution of the intracluster medium, and
therefore it can be used to estimate the cluster magnetic field
strength along the line of sight.

The RM observed in radio galaxies may not be all due to the cluster
magnetic field if the RM gets locally enhanced by the intracluster
medium compression due to the motion of the radio galaxy
itself. However a statistical RM investigation of point sources
(Clarke et
al. 2001,
Clarke 2004)
shows a clear broadening of the RM
distribution toward small projected distances from the cluster centre,
indicating that most of the RM contribution comes from the
intracluster medium. This study included background sources, which
showed similar enhancements as the embedded sources.

Figure 6. RM magnitudes of a sample of
radio galaxies located in cooling flow clusters, plotted as a function
of the cooling flow rate X (from
[Taylor et
al. 2002]).

Dolag et
al. (2001b)
showed that, in the framework of hierarchical cluster
formation, the correlation between two observable parameters, the RM
and the cluster X-ray surface brightness in the source location, is
expected to reflect a correlation between the cluster magnetic field
and gas density. Therefore, from the analysis of the RM versus X-ray
brightness it is possible to infer the trend of magnetic field versus
gas density.

On the basis of the available high quality RM images, increasing
attention is given to the power spectrum of the intracluster magnetic
field fluctuations. Several studies
(Enßlin
& Vogt 2003,
Murgia et al. 2004)
have shown that detailed RM images of radio galaxies can be used to infer
not only the cluster magnetic field strength, but also the cluster
magnetic field power spectrum. The analyses of Vogt &
Enßlin
(2003,
2005)
and
Guidetti et
al. (2007)
suggest that the power spectrum is of the
Kolmogorov type, if the auto-correlation length of the magnetic
fluctuations is of the order of few kpc. However,
Murgia et
al. (2004)
and
Govoni et
al. (2006)
pointed out that shallower magnetic field power spectra are possible if
the magnetic field fluctuations extend out to several tens of kpc.

As shown in Table 3 of
Govoni &
Feretti (2004),
the different methods
available to measure intracluster magnetic fields show quite
discrepant results (even more than a factor 10). RM estimates are
about an order of magnitude higher than the measures derived both from
the synchrotron diffuse radio emission and the non-thermal hard X-ray
emission (~ 1 - 5 µG vs. ~ 0.2 - 1 µG).

This can be due to several factors. Firstly, equipartition values are
severely affected by the already mentioned physical assumptions of
this method. Secondly, while RM estimates give a weighted average of
the field along the line of sight, equipartition and Compton
scattering measures are made by averaging over larger
volumes. Additionally, discrepancies can be due to spatial profiles of
both the magnetic field and the gas density not being constant all
over the cluster
(Goldshmidt
& Rephaeli 1993),
or due to compressions,
fluctuations and inhomogeneities in the gas and in the magnetic field,
related to the presence of radio galaxies or to the dynamical history
of the cluster (e.g. on-going merging events)
(Beck et al. 2003,
Rudnick &
Blundell 2003b,
Johnston-Hollitt
2004).
Finally, a proper modelling of the Compton
scattering method should include a) the effects of aged electron
spectra, b) the expected radial profile of the magnetic field, and c)
possible anisotropies in the pitch angle distribution of electrons
(Brunetti et
al. 2001,
Petrosian
2001).

An additional method of estimating cluster magnetic fields comes from
the X-ray analysis of cold fronts
(Vikhlinin et
al. 2001).
These X-ray cluster features, discovered by
Markevitch et
al. (2000)
thanks to the exquisite spatial resolution of the Chandra
satellite, result
from dense cool gas moving with near-sonic velocities through the less
dense and hotter ICM. Cold fronts are thus subject to Kelvin-Helmholtz
(K-H) instability that, for typical cluster and cold front properties
(Mach number, gas temperatures, cluster-scale length), could quickly
disturb the front outside a narrow
( 10°) sector in
the direction of the cool cloud motion
5.
Through the Chandra observation of A 3667,
Vikhlinin et
al. (2001)
instead revealed a cold front that is stable within a ± 30°
sector. They showed that a ~ 10 µG magnetic field oriented
nearly parallel to the front is able
to suppress K-H instability, thus preserving the front structure, in a
± 30° sector. The estimated magnetic field value,
significantly higher than the typical measures given by the other
methods outside cluster cooling flows, is likely an upper limit of the
absolute field strength. Near the cold front the field is actually
amplified by tangential gas motions (see
Vikhlinin et
al. 2001).

Variations of the magnetic field structure and strength with the
cluster radius have been recently pointed out by
Govoni et
al. (2006).
By combining detailed multi-wavelength and numerical studies we will get
more insight into the strength and structure of intracluster magnetic
fields, and into their connection with the thermodynamical evolution
of galaxy clusters. More detailed comparisons of the different
approaches for measuring intracluster magnetic fields can be found,
for instance, in
Petrosian
(2003)
and
Govoni &
Feretti (2004).